行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}1 & {-}1 & {-}1 & 1 & 2 \\ {-}1 & {-}1 & 1 & {-}2 & 2 \\ 2 & 0 & 2 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 & 1 & 2 \\ 0 & 1 & {-}1 & 2 & 1\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 & 2 \\ {-}1 & {-}1 & 1 & {-}2 & 2 \\ 2 & 0 & 2 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 & 1 & 2 \\ 0 & 1 & {-}1 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第5列で展開する : [2, 2, -1, 2, 1]

\( \qquad\quad = (-1)^{5-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| + (-1)^{5-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| + (-1)^{5-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| + (-1)^{5-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| + (-1)^{5-5}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{5-1}\times2\times(-7) + (-1)^{5-2}\times2\times33 + (-1)^{5-3}\times(-1)\times(-9) + (-1)^{5-4}\times2\times(-4) + (-1)^{5-5}\times1\times(-36) \)

\( \qquad\quad = -99 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, -1, 1, 2]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 0 & 2 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times(-6) + (-1)^{4-2}\times(-1)\times1 + (-1)^{4-3}\times1\times2 + (-1)^{4-4}\times2\times4 \)

\( \qquad\quad = -7 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 0 & 2 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times(-2) + (-1)^{3-2}\times2\times2 + (-1)^{3-3}\times(-1)\times(-2) \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-2) + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-1)\times(-1) \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times2 + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-1)\times2 \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-2) + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [1, -1, 1, 2]

\( \qquad\quad = (-1)^{4-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 0 & 2 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times1\times(-6) + (-1)^{4-2}\times(-1)\times3 + (-1)^{4-3}\times1\times(-6) + (-1)^{4-4}\times2\times12 \)

\( \qquad\quad = 33 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 0 & 2 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times(-2) + (-1)^{3-2}\times2\times2 + (-1)^{3-3}\times(-1)\times(-2) \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-2) + (-1)^{3-2}\times2\times1 + (-1)^{3-3}\times(-1)\times(-3) \)

\( \qquad\quad = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times2 + (-1)^{3-2}\times2\times1 + (-1)^{3-3}\times(-1)\times2 \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-2) + (-1)^{3-2}\times2\times(-3) + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 12 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ {-}2 & {-}1 & 2 & 1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [1, -2, 1, 2]

\( \qquad\quad = (-1)^{4-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times1\times1 + (-1)^{4-2}\times(-2)\times3 + (-1)^{4-3}\times1\times2 + (-1)^{4-4}\times2\times0 \)

\( \qquad\quad = -9 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-2) + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-1)\times(-1) \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}2 & {-}1 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-2) + (-1)^{3-2}\times2\times1 + (-1)^{3-3}\times(-1)\times(-3) \)

\( \qquad\quad = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}1 \\ 0 & 1\end{array}\,\right| = (-2)\times1 - (-1)\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-1) + (-1)^{3-2}\times1\times1 + (-1)^{3-3}\times(-1)\times(-2) \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-1) + (-1)^{3-2}\times1\times(-3) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ 0 & 1 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [1, -2, -1, 2]

\( \qquad\quad = (-1)^{4-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 0 & 1 & {-}1\end{array}\,\right| + (-1)^{4-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 2 & 0 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times1\times2 + (-1)^{4-2}\times(-2)\times(-6) + (-1)^{4-3}\times(-1)\times2 + (-1)^{4-4}\times2\times(-8) \)

\( \qquad\quad = -4 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times2 + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-1)\times2 \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times2 + (-1)^{3-2}\times2\times1 + (-1)^{3-3}\times(-1)\times2 \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ 0 & 1\end{array}\,\right| = 2\times1 - 0\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 0 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, -1]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-1) + (-1)^{3-2}\times1\times1 + (-1)^{3-3}\times(-1)\times(-2) \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 0 & 1\end{array}\,\right| = 1\times1 - (-1)\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 2 & 0 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times2 + (-1)^{3-2}\times1\times2 + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 & 1 \\ {-}1 & {-}1 & 1 & {-}2 \\ 2 & 0 & 2 & {-}1 \\ {-}2 & {-}1 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [1, -2, -1, 1]

\( \qquad\quad = (-1)^{4-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| + (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| + (-1)^{4-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2\end{array}\,\right| + (-1)^{4-4}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 2 & 0 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times1\times4 + (-1)^{4-2}\times(-2)\times12 + (-1)^{4-3}\times(-1)\times0 + (-1)^{4-4}\times1\times(-8) \)

\( \qquad\quad = -36 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-2) + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ 2 & 0 & 2 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-2) + (-1)^{3-2}\times2\times(-3) + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 12 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 0 \\ {-}2 & {-}1\end{array}\,\right| = 2\times(-1) - 0\times(-2) = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ {-}2 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-1) + (-1)^{3-2}\times1\times(-3) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times(-2) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}2 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-2) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}1 & {-}1 \\ {-}1 & {-}1 & 1 \\ 2 & 0 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times2 + (-1)^{3-2}\times1\times2 + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ 2 & 0\end{array}\,\right| = 1\times0 - (-1)\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}1 \\ {-}1 & {-}1\end{array}\,\right| = 1\times(-1) - (-1)\times(-1) = -2 \)